The present invention relates to ultrawide bandwidth (UWB) transmitters, receivers and transmission schemes. More particularly, the present invention relates to a method and system for sending data across a UWB signal using M-ary bi-orthogonal keying.
The following is a general description of a UWB system, noting particularly how it is applicable to wireless networks. Although UWB technology has also been used in radar and ranging applications, the following discussion addresses only issues relevant to wireless networking applications.
It is helpful to briefly note some important design issues for indoor wireless networks. Such systems will need to operate over relatively short ranges in environments with multipath interference, but will need to provide high data rates, preferably using spectrum licensed by the Federal Communications Commission (FCC). Also, such systems are often used to support mobility, so they need low power dissipation to enable battery operation and, as always, low cost and complexity is an advantage.
Characteristics of UWB Systems
One embodiment of a UWB system uses signals that are based on trains of short duration pulses (also called chips) formed using a single basic pulse shape. The interval between individual pulses can be uniform or variable, and there are a number of different methods that can be used for modulating the pulse train with data for communications. One common characteristic in this embodiment, however, is that the pulse train is transmitted without translation to a higher carrier frequency, and so UWB transmissions using these sorts of pulses are sometimes also termed “carrier-less” radio transmissions. In other words, in this embodiment a UWB system drives its antenna directly with a baseband signal.
Another important point common to UWB systems is that the individual pulses are very short in duration, typically much shorter than the interval corresponding to a single bit, which can offer advantages in resolving multipath components. We can represent a general UWB pulse train signal as a sum of pulses shifted in time, as shown in Equation 1:
                              s          ⁡                      (            t            )                          =                              ∑                          k              =                              -                ∞                                      ∞                    ⁢                                    a              k                        ⁢                          p              ⁡                              (                                  t                  -                                      t                    k                                                  )                                                                        (        1        )            
Here s(t) is the UWB signal, p(t) is the basic pulse shape, and ak and tk are the amplitude and time offset for each individual pulse. Because of the short duration of the pulses, the spectrum of the UWB signal can be several gigahertz or more in bandwidth. An example of a typical pulse stream is shown in FIG. 1. Here the pulse is a Gaussian monopulse with a peak-to-peak time (Tp-p) of a fraction of a nanosecond, a pulse period Tp of several nanoseconds, and a bandwidth of several gigahertz.
UWB Systems Limited to Low Power Spectral Density
UWB systems in general have extremely wide absolute bandwidth relative to most existing wireless systems. This bandwidth is a direct consequence of the use of sub-nanosecond pulses that leads to signal bandwidths of several gigahertz or more. Because these signals are also transmitted without translation to higher center frequencies, it is clear that these signals will occupy the same frequency bands that are already in use by many existing spectrum users.
Because of rulings by the FCC, future UWB systems will likely be limited to operations using extremely low power spectral density (as measured in dBm/MHz). Based on this fact, it is clear that even with a bandwidth of several gigahertz, UWB systems will also be limited to relatively low total transmit power. For example, a UWB system with 5 GHz of bandwidth might have a maximum total transmit power of only a small fraction of a milliwatt over the entire 5 GHz of bandwidth.
Operation in the Power-Limited Regime
The bandwidth efficiency of a digital modulation scheme that transmits B bits in T seconds (R bits/sec) using a bandwidth of W hertz is given by R/W=B/(WT) bits/s/Hz. As we will see, the bandwidth efficiency of a UWB system is not important in the sense of how efficiently it uses spectrum, but rather the value of this ratio serves to distinguish UWB systems from more typical narrowband systems. Based on this ratio, R/W, digital communications systems can be classified as operating in either the bandwidth-limited regime or the power-limited regime of the bandwidth-efficiency plane. This classification has fundamental implications for many of the important trade-offs that must be made in the design of efficient communications systems.
For future UWB systems, the R/W ratio will likely be very low for the system to have any useful range. For example, even for a relative high-rate wireless network (say 100 Mbps), the bandwidth efficiency of a UWB wireless network will be as low as 1/20 or even 1/50, depending on the bandwidth W. The primary consequence of this low value for the ratio R/W is that UWB systems will almost certainly operate well within the power-limited regime of the bandwidth-efficiency plane.
The Critical Importance of Power Efficiency
The main result of UWB operation in the power-limited regime is that such systems will be very sensitive to design issues that affect the power efficiency of the system. For this reason, the analysis in the following sections will focus on the critical issues of power efficiency of the UWB modulation techniques, as well as the spectral effects of modulation that might also affect allowable transmit power levels. The implications of power-limited operation will also influence system-level trade-offs between range and data rate, as well as trade-offs between complexity and performance in the form of forward error-correction.
Multipath Robustness and Precision Ranging
One frequently mentioned benefit of ultra-wide bandwidth is a robustness to the effects of multipath interference. Multipath interference results when multiple time-displaced copies of a signal reach a receiver at the same time because of signal bounces in a cluttered environment. This robustness is a result of two distinct factors: (1) wide fractional bandwidth leads to less severe multipath fading, which is particularly important for low-power wireless systems; and (2) wide absolute bandwidth enables resolution of multipath components and constructive use of multipath.
The effect of reduced multipath fading can be partially understood from a frequency-domain perspective by realizing that the absolute signal bandwidth of the UWB signal is much greater than the coherence bandwidth of nearly any conceivable multipath channel. Any frequency-selective fades due to multipath will only affect a small portion of the signal power for any channel realization. Previous work provides empirical evidence that UWB signals experience a much lower variance in received signal power in the presence of multipath than do narrowband signals.
For UWB signals, robustness to multipath fading is a result not just of the wide system bandwidth, however, but is also a result of the large ratio of system bandwidth to center frequency, i.e., the fractional bandwidth. A large fractional bandwidth means that there is a corresponding large variation in the mode and degree of RF energy interaction with the surrounding environment over the entire UWB bandwidth. Environmental interactions such as scattering, refraction and reflection depend on the wavelength of the RF signals, and so the large fractional bandwidth leads to relatively low correlation in the fading properties of the different regions of the UWB bandwidth. Thus, the properties of UWB signals should lead to more robust multipath performance even than systems with equal bandwidth but much higher center frequencies (i.e. lower fractional bandwidths).
The wide absolute bandwidth of UWB signals also provides fine time resolution that enables a receiver to resolve and combine individual multipath components, avoiding destructive interference.
Analysis of UWB Modulation Choices
Under current FCC regulations, UWB transmit power is limited by the power spectral density (PSD) of the transmitted signal. FIG. 2 is a graph showing the power spectral density limits currently put in force by the FCC.
This limitation affects the selection of a UWB modulation scheme in two distinct ways. First, the modulation technique needs to be power efficient. In other words, the modulation needs to provide the best error performance for a given energy per bit. Second, the choice of a modulation scheme affects the structure of the PSD in the sense that it affects the distribution of signal power over different frequency bands. If a particular modulation scheme results in the concentration of signal power in narrow frequency ranges, it has the potential to impose additional constraints on the total transmit power in order to satisfy the PSD limitations.
As we compare different modulation schemes, we therefore examine both the power efficiency and the effect of the modulation on the PSD. In the sections that follow, we examine a number of modulation schemes that have been proposed for UWB, including several forms of pulse amplitude modulation (PAM), such as: positive pulse amplitude modulation (PPAM), on-off keying (OOK), and binary phase-shift keying (BPSK), as well as pulse-position modulation (PPM).
Pulse Amplitude Modulation
As noted above, one general form of a UWB signal is a simple pulse train. Assuming that pulses are uniformly spaced in time (i.e. the kth pulse occurs at time t=kT), then we can simplify Equation (1) to:
                              s          ⁡                      (            t            )                          =                              ∑                          k              =                              -                ∞                                      ∞                    ⁢                                    a              k                        ⁢                          p              ⁡                              (                                  t                  -                  kT                                )                                                                        (        2        )            
where T is the pulse-spacing interval. From this general form of PAM, we can analyze several specific modulation techniques by choosing the mapping from data bits to pulse weights (ak) in different ways. These different techniques are illustrated in FIGS. 3A-3C, and are described in the following paragraphs.
FIGS. 3A-3C are graphs showing exemplary pulse streams for OOK, PPAM, and BPSK modulation schemes, respectively. In each case, they show a data sequence “1 0 1 0.” FIGS. 4A-4C are constellation diagrams for the modulation schemes of FIGS. 3A-3C, respectively. As shown in FIGS. 4A-4C, the constellation diagrams for OOK, PPAM, and BPSK are all one-dimensional, differing only in the symbol constellation's position relative to the origin.
On-Off Keying
As shown in FIG. 3A, OOK defines the data by the presence or absence of a pulse. A “1” is indicated by a pulse, and a “0” is indicated by the absence of a pulse. Thus, the bit stream “1 0 1 0” is indicated by the sequence of: a pulse, a blank where a pulse should be, a pulse, and another blank.
This embodiment has akε{0,2}, i.e., data bits are transmitted by either the presence or absence of a pulse at time t=tk. In the constellation diagram in FIG. 4A, this results in symbol points at (0,0) and (2,0).
Positive Pulse Amplitude Modulation
As shown in FIG. 3B, PPAM defines the data by the amplitude of the pulse. A “1” is indicated by a large pulse, and a “0” is indicated by a small pulse. Thus, the bit stream “1 0 1 0” is indicated by the sequence of: a large pulse, a small pulse, a large pulse, and a small pulse.
This embodiment uses strictly positive values for the two pulse weights, so that akε{α0,α1} where 0<α0<α1. This corresponds to transmitting either a large or small amplitude pulse based on the value of the source bit. In the constellation diagram of FIG. 4B this is shown as having signal points at (α0, 0) and (α1, 0).
Binary Phase Shift Keying
As shown in FIG. 3C, BPSK defines the data by the polarity of the pulse. A “1” is indicated by a non-inverted pulse, and a “0” is indicated by an inverted pulse. Thus, the bit stream “1 0 1 0” is indicated by the sequence of: a non-inverted pulse, an inverted pulse, a non-inverted pulse, and an inverted pulse.
In this embodiment akε{−1,+1}. This corresponds to transmitting either a non-inverted or an inverted pulse based on the value of the source bit. In the constellation diagram of FIG. 4C this is shown as having signal points at (−1, 0) and (1, 0).
Pulse-Position Modulation
One other technique proposed for UWB pulse modulation, PPM, is fundamentally different from the PAM techniques described above because the pulses are not uniformly spaced in time. Rather, the source data bits are used to modulate the time position of the individual pulses instead of the pulse amplitudes. For example, binary PPM encodes the data bits in the pulse stream by advancing or delaying individual pulses in time relative to uniform reference positions. In this case, the equation for the UWB signal becomes
                              s          ⁡                      (            t            )                          =                                            ∑                              k                =                                  -                  ∞                                            ∞                        ⁢                          p              ⁡                              (                                  t                  -                                      t                    k                                                  )                                              =                                    ∑                              k                =                                  -                  ∞                                            ∞                        ⁢                          p              ⁡                              (                                  t                  -                  kT                  +                                                            a                      k                                        ⁢                    β                                                  )                                                                        (        3        )            
Here the data bits are mapped to the direction of the time shifts, ak, where akε{−1,1}, and β is the amount of pulse advance or delay in time relative to the reference (unmodulated) position. When we consider the constellation diagram for binary PPM, we find that the plot is no longer one dimensional, as it is for the binary PAM techniques. For PPM, the presence of two pulse with different time offsets results in a two-dimensional constellation plot. To find the specific location of the symbol points within the plot, however, we need to determine the correlation ρ between the two different symbols, the advanced and delayed pulses.
                    ρ        =                                            ∫                              =                                  -                  ∞                                            ∞                        ⁢                                          p                ⁡                                  (                                      t                    -                                          β                      ⁢                                                                                          ⁢                      T                                                        )                                            ⁢                              p                ⁡                                  (                                      t                    +                                          β                      ⁢                                                                                          ⁢                      T                                                        )                                            ⁢                                                          ⁢                              ⅆ                t                                                                        ∫                              =                                  -                  ∞                                            ∞                        ⁢                                          p                ⁡                                  (                  t                  )                                            ⁢                              p                ⁡                                  (                  t                  )                                            ⁢                              ⅆ                t                                                                        (        4        )            
FIGS. 5A-5C are constellation diagrams for pulse position modulation schemes under various conditions for binary PPM based on the pulse shown in FIG. 1. FIG. 5A shows as situation where the pulses are orthogonal (i.e., ρ=0); FIG. 5B shows the situation where the pulses are not orthogonal and ρ>0; and FIG. 5C shows the situation where the pulses are not orthogonal and ρ<0.
For the non-orthogonal cases of binary PPM, the orthogonal basis function used to define the constellation plot can be found using Gram-Schmidt orthogonalization for the two non-orthogonal pulses. The constellation diagrams in FIGS. 5A-5C all have symbol points at (1,0) and (ρ,√{square root over (1−ρ2)}), and the two symbol points lie on the unit circle (when normalized to unit energy).
In the case where the two different locations of the pulse have no overlap in time, the correlation will clearly be (ρ≈0) and the binary PPM becomes orthogonal modulation. The constellation for this case is shown in FIG. 5A, where the symbol points are (1,0) and (0,1). Here the two orthogonal pulses have been used to create orthogonal basis vectors for the constellation plot.
When the two pulses overlap, the correlation ρ in general will not be zero, but will range between one and some minimum (possibly negative) value.
Comparison of Power Efficiency for Binary Modulation
We can use the constellation diagrams in FIGS. 4A-4C and 5A-5C to compare the power efficiency of the various binary modulation techniques by computing the inter-symbol distance, d, as a function of average symbol energy, Es. For OOK, we have
            E      s        =                  (                              0            2                    +                      d            2                          )            2        ,            so      ⁢                                        ⁢                                      ⁢      d        =                            2          ⁢                      E            s                              .      For positive-valued PAM (PPAM) we see that
      d    =          (                        α          1                -                  α          0                    )        ,            so      ⁢                          ⁢              E        s              =                            (                                    α              0              2                        +                                          (                                  d                  +                                      α                    0                                                  )                            2                                      2            .      Solving for d, we get d=(√{square root over (2Es−α02)}−α0). If we assume α0≧0, then we have d≦√{square root over (2ES)}, which is satisfied with equality when α0=0 (i.e. when PPAM becomes OOK). For antipodal binary PAM (BPSK) we have
            E      s        =                  (                  d          2                )            2        ,            so      ⁢                          ⁢      d        =          2      ⁢                                    E            s                          .            
For binary PPM, the inter-symbol distance depends on the correlation between the advanced and delayed pulses defined in Equation (4) and for the general case, d=√{square root over (2ES(1−ρ))}. Here we see that if the value of ρ ranges between −1 and +1, the distance can range between d=0 and d=2√{square root over (Es)}. The actual maximum and minimum values for ρ that determine this range of possible inter-symbol distances depend on the specific shape of the pulse p(t) and can be determined according to Equation (4) for different values of β. For the example Gaussian monopulse shown in FIG. 1, the value of ρ as defined in Equation (4) ranges from (+1) to approximately (−0.45) as β ranges from zero to several multiples of Tp.
TABLE 1Differences Between Modulation TechniquesPower EfficiencyModulationInter-symbolRelative toClassSpecific FormDistanceAntipodal SignalingPulse-Orthogonal  d  =            2      ⁢              E        b            −3 dBpositionNon-orthogonal  d  =            2      ⁢                        E          b                ⁡                  (                      1            -            ρ                    )                    <1.4 dB (variable)ModulationAmplitudePositive PAM  d  <            2      ⁢              E        b            <−3 dB ModulationOOK  d  =            2      ⁢              E        b            −3 dBAntipodal  d  =      2    ⁢                  E        b             0 dB
These results show significant differences between the modulation techniques and are summarized in Table 1. The orthogonal PPM and OOK techniques are equally efficient and the positive PAM system is less so, but becomes the same in the limit as the PAM becomes OOK. Non-orthogonal PPM has a power efficiency that depends on the symbol correlation ρ, but is still suboptimal. Antipodal signaling (BPSK) provides the greatest inter-symbol distance for a given average symbol energy. This difference provides at least a 3 dB advantage in efficiency relative to OOK, PPAM, or orthogonal PPM, and to achieve the same bit error rate (which is a function of distance) PPM or OOK must use double the bit energy, or 3 dB higher Eb.
Decomposition of Binary Modulation Techniques
For the binary PAM techniques depicted in FIGS. 4A-4C, the constellation diagrams differ only in their position relative to the origin. It is a well-known result in communications theory that power efficiency depends on the mean of the symbol constellation—this is why the zero-mean property of the BPSK makes it superior in the ratio of inter-symbol distance to symbol energy. Another way to understand this difference is to decompose the weight sequence into a constant value sequence added to a zero-mean random sequence: ak=μa+zk. This sequence decomposition allows us to represent the UWB pulse train as the sum of an unmodulated component pulse train and an antipodal component pulse train:
                              s          ⁡                      (            t            )                          =                                            ∑                              k                =                                  -                  ∞                                            ∞                        ⁢                                          μ                a                            ⁢                              p                ⁡                                  (                                      t                    -                    kT                                    )                                                              =                                    ∑                              k                =                                  -                  ∞                                            ∞                        ⁢                                          z                k                            ⁢                              p                ⁡                                  (                                      t                    -                    kT                                    )                                                                                        (        5        )            
From this result we can easily see the source of the difference in power efficiency for the PAM techniques. The energy in the unmodulated component of the pulse train above does not contribute to communicating data between the transmitter and receiver, and is effectively wasted. Only the energy in the antipodal component contributes to the communications process. The greater the energy in the unmodulated component (i.e. the higher the mean μa for a give distance d) the poorer is the power efficiency of the modulation. BPSK is thus seen to be optimal for binary techniques since it has zero-mean and all of its energy is contained in the antipodal component of the pulse train.
For PPM, we can perform a similar, but more general, decomposition of the pulse train. Here we must use unmodulated and antipodal components that, unlike PAM, have different pulse shapes. We define two new pulses:
                                          m            ⁡                          (              t              )                                =                                                    p                ⁡                                  (                                      t                    -                                          β                      ⁢                                                                                          ⁢                      T                                                        )                                            +                              p                ⁡                                  (                                      t                    +                                          β                      ⁢                                                                                          ⁢                      T                                                        )                                                      2                          ,                              and            ⁢                                                  ⁢                          b              ⁡                              (                t                )                                              =                                                    p                ⁡                                  (                                      t                    -                                          β                      ⁢                                                                                          ⁢                      T                                                        )                                            -                              p                ⁡                                  (                                      t                    +                                          β                      ⁢                                                                                          ⁢                      T                                                        )                                                      2                                              (        6        )            
These pulses represent the unmodulated [m(t)] and antipodal [b(t)] pulse train components. We can use these two pulses to write the UWB pulse train as the sum of two separate component pulse trains:
                              s          ⁡                      (            t            )                          =                                            ∑                              k                =                                  -                  ∞                                            ∞                        ⁢                                          μ                a                            ⁢                              m                ⁡                                  (                                      t                    -                    kT                                    )                                                              +                                    ∑                              k                =                                  -                  ∞                                            ∞                        ⁢                                          z                k                            ⁢                              b                ⁡                                  (                                      t                    -                    kT                                    )                                                                                        (        7        )            
Using this decomposition, we see that data bits are transmitted by sending either [m(t)+b(t)] or [m(t)−b(t)] at each time interval t=kT. The sign of the component m(t) is independent of the data value and is therefore not modulated. Two examples of this decomposition for binary PPM are shown in FIGS. 6A and 6B for values of β that result in both overlapping and non-overlapping pulses.
FIGS. 6A-6D are graphs showing component pulses for the decomposition of binary PPM into unmodulated and antipodal pulse trains. FIG. 6A shows the original pulses with β=5Tp; FIG. 6B shows the original pulses with β=1.5Tp; FIG. 6C shows the unmodulated component pulse [m(t)] and antipodal component pulse [b(t)] for β=5Tp; and FIG. 6D shows the unmodulated component pulse [m(t)] and antipodal component pulse [b(t)] for β=1.5Tp.
As with the PAM cases above, we can see that the energy in the unmodulated component of the pulse train defined by m(t) is useless in the communication of information and leads to inefficient modulation.
Spectral Effects of Modulation Techniques
Another important consideration in evaluating a UWB modulation technique is the effect of the modulation on the spectrum of the transmitted signal. As noted in an earlier section, UWB signals have been limited by the FCC by the peak of their PSD, so that for best system performance signals should be designed to maximize transmit power for given limits on PSD levels.
Spectral Analysis for PAM
To understand the effect of the modulation scheme on the UWB signal, we need to find the spectrum not of the isolated pulse, but of the modulated pulse train. If we assume that the modulating data are random, the transmitted pulse train is also a random signal and as such does not have a deterministic Fourier transform. However, we can still understand the effects of modulation on the spectral distribution of signal power by finding its expectation over the random source data sequences. This power spectral density (PSD) of the transmit signal, s(t), is the Fourier transform of the signal autocorrelation and is denoted by ΦSS(ƒ). Because the pulses in a PAM UWB signal are uniformly spaced as in Equation (2), we can derive a general form for the PSD of the PAM signals as follows:ΦSS(ƒ)=|P(ƒ)|2Φaa(ƒ)  (8)
Here P(f) is the Fourier transform of the basic pulse, p(t), and Φaa(ƒ) is the PSD of the random data sequence, ak, which is hereafter assumed to be a wide-sense stationary random sequence. If we assume that the pulse weights ak correspond to the data bits to be transmitted and that the random data are independent and identically distributed (IID), then the PSD can be determined as follows:
                                          Φ            aa                    ⁡                      (            f            )                          =                              σ            a            2                    +                                                    μ                a                2                            T                        ⁢                                          ∑                                  k                  =                                      -                    ∞                                                  ∞                            ⁢                              δ                ⁡                                  (                                      f                    -                                          k                      T                                                        )                                                                                        (        9        )            
where σa2 and μa are the variance and mean of the weight sequence and δ(f) is a unit impulse function. This PSD is periodic in the frequency domain with period
  f  =      1    T  because it is the transform of the discrete auto-correlation sequence, Φaa(k)=E{an+kan*}. This PSD in Equation (9) has both a continuous portion and discrete spectral lines, corresponding to the first and second terms on the right-hand side. It is worth noting that the magnitude of the spectral lines depends on the mean of the weights, μa. In light of the decomposition described above, we see that the energy in the unmodulated component of the pulse train is the energy in the spectral lines and the energy in the antipodal component is the energy of the continuous spectral component.
When we combine the results of Equations (8) and (9) we see that the resulting PSD of the transmitted signal is equivalent to the result of filtering a weighted impulse sequence through a filter with frequency response P(f):
                                          Φ            ss                    ⁡                      (            f            )                          =                                                            σ                a                2                            T                        ⁢                                                                            P                  ⁡                                      (                    f                    )                                                                              2                                +                                                    μ                a                2                                            T                2                                      ⁢                                          ∑                                  k                  =                                      -                    ∞                                                  ∞                            ⁢                                                                                                              P                      ⁡                                              (                                                  k                          T                                                )                                                                                                  2                                ⁢                                  δ                  ⁡                                      (                                          f                      -                                              k                        T                                                              )                                                                                                          (        10        )            
At this point we can again consider the different modulation techniques for PAM described earlier. The PSD for the OOK signal with pulse amplitudes weights akε{0,2} is determined as follows:
                                          Φ                          ss              ,              OOK                                ⁡                      (            f            )                          =                                            1              T                        ⁢                                                                            P                  ⁡                                      (                    f                    )                                                                              2                                +                                    1                              T                2                                      ⁢                                          ∑                                  k                  =                                      -                    ∞                                                  ∞                            ⁢                                                                                                              P                      ⁡                                              (                                                  k                          T                                                )                                                                                                  2                                ⁢                                  δ                  ⁡                                      (                                          f                      -                                              k                        T                                                              )                                                                                                          (        11        )            
In this equation we see that OOK results in discrete spectral lines in the PSD of the UWB signal. The spectral lines are spaced at a frequency interval of
  f  =      1    T  and each line has power proportional to P(f) evaluated at
  F  =            k      T        .  For OOK the total power in the spectral lines is equal to the power in the continuous component of the PSD, as shown above. A similar result is obtained for the positive-valued PAM signal, where we have
      σ    a    2    =                                          (                                          α                0                            -                              α                1                                      )                    2                4            ⁢                          ⁢      and      ⁢                          ⁢              μ        a              =                            (                                    α              0                        +                          α              1                                )                2            .      Substituting these values in Equation (8) the PSD becomes:
                                          Φ                          ss              ,              PPAM                                ⁡                      (            f            )                          =                                                                              (                                                            α                      0                                        -                                          α                      1                                                        )                                2                                            4                ⁢                T                                      ⁢                                                                            P                  ⁡                                      (                    f                    )                                                                              2                                +                                                                      (                                                            α                      0                                        -                                          α                      1                                                        )                                2                                            4                ⁢                                  T                  2                                                      ⁢                                          ∑                                  k                  =                                      -                    ∞                                                  ∞                            ⁢                                                                                                              P                      ⁡                                              (                                                  k                          T                                                )                                                                                                  2                                ⁢                                  δ                  ⁡                                      (                                          f                      -                                              k                        T                                                              )                                                                                                          (        12        )            
We see that, as with OOK, there are spectral lines present in the transmitted signal for positive-valued PAM and furthermore that the magnitude of the lines increases with the weight sequence mean. Note that PPAM spectrum becomes the same as the OOK spectrum when α0→0.
The situation is very different for antipodal signaling, where akε{−1,+1}, so that σa2=1 and μa=0. In this case, the PSD becomes simply:
                                          Φ                          ss              ,              BPSK                                ⁡                      (            f            )                          =                                                            σ                a                2                            T                        ⁢                                                                            P                  ⁡                                      (                    f                    )                                                                              2                                =                                    1              T                        ⁢                                                                            P                  ⁡                                      (                    f                    )                                                                              2                                                          (        13        )            
Here we see that the spectral lines vanish because of the zero mean of the weight sequence. Because the PSD for BPSK has no lines, the spectral distribution of energy does not depend on the pulse interval T or the pulse-repetition frequency (PRF). Rather the presence of T in Equation (13) only shows that the total power of the transmit signal increases linearly at all frequencies with the PRF when pulse amplitude is constant.
Spectral Analysis for PPM
The results of Equations (8) and (9) do not directly apply to the case of PPM because the pulses do not have uniform spacing in time. To find the PSD for PPM signals, however, we can use the decomposition technique described in Equation (7) above that allowed us to represent the PPM signal as the sum of two uniformly spaced pulse trains. From the definitions in Equation (6) it is clear that m(t) and b(t) are orthogonal regardless of the orthogonality of the shifted pulses p(t−β) and p(tβ). Using this fact, we can find the PSD of the composite pulse train in Equation (7), the PSD of the binary PPM signal as follows:
                                          Φ                          ss              ,              PPM                                ⁡                      (            f            )                          =                                                            σ                a                2                            T                        ⁢                                                                            B                  ⁡                                      (                    f                    )                                                                              2                                +                                                    μ                a                2                                            T                2                                      ⁢                                          ∑                                  k                  =                                      -                    ∞                                                  ∞                            ⁢                                                                                                              M                      ⁡                                              (                                                  k                          T                                                )                                                                                                  2                                ⁢                                  δ                  ⁡                                      (                                          f                      -                                              k                        T                                                              )                                                                                                          (        14        )            
Where B(f) and M(f) are the Fourier transforms of the component pulses b(t) and m(t), respectively. As with the case of the PAM signals, it is clear that the energy that corresponded to the unmodulated pulse train in Equation (7) here translates to energy contained in spectral lines. Similarly, the energy in the antipodal portion of the signal translates to the energy of the continuous spectral component of Equation (14).
One significant difference between the PAM and PPM spectra is that for PPM the envelope of the magnitudes of the spectral lines can be different from the shape of the continuous spectrum.
The continuous component of the PSD has a shape that depends on B(f), but the power distribution in the spectral lines depends on M(f). These spectral lines still have a frequency spacing of
      f    =          1      T        ,but the distribution of power in the lines can be significantly different.
In general, both the distribution of energy between the discrete and continuous components of the spectrum, as well as the distribution of spectral energy with respect to frequency, depend on the shape of the original pulse p(t) and the magnitude of the time shift, βT. As with the PAM signals, we can conclude that from the viewpoint of the system designer it is desirable to minimize the energy in the spectral lines. For PPM this is done by minimizing the correlation value ρ, with the additional consideration that the shape of the component pulses m(t) and b(t) may result in less uniform distribution of energy in the spectrum. This could in turn lead to suboptimal designs for a PSD-limited system.